# A Hilbert space approach to effective resistance metric - Mathematics > Functional Analysis

Abstract: A resistance network is a connected graph $G,c$. The conductance function$c {xy}$ weights the edges, which are then interpreted as conductors ofpossibly varying strengths. The Dirichlet energy form $\mathcal E$ produces aHilbert space structure which we call the energy space ${\mathcal H} {\mathcalE}$ on the space of functions of finite energy.We use the reproducing kernel $\{v x\}$ constructed in \cite{DGG} to analyzethe effective resistance $R$, which is a natural metric for such a network. Itis known that when $G,c$ supports nonconstant harmonic functions of finiteenergy, the effective resistance metric is not unique. The two most naturalchoices for $Rx,y$ are the free resistance- $R^F$, and the wiredresistance- $R^W$. We define $R^F$ and $R^W$ in terms of the functions $v x$and certain projections of them. This provides a way to express $R^F$ and$R^W$ as norms of certain operators, and explain $R^F eq R^W$ in terms ofNeumann vs. Dirichlet boundary conditions. We show that the metric space$G,R^F$ embeds isometrically into ${\mathcal H} {\mathcal E}$, and the metricspace $G,R^W$ embeds isometrically into the closure of the space of finitelysupported functions; a subspace of ${\mathcal H} {\mathcal E}$.Typically, $R^F$ and $R^W$ are computed as limits of restrictions to finitesubnetworks. A third formulation $R^{tr}$ is given in terms of the trace of theDirichlet form $\mathcal E$ to finite subnetworks. A probabilistic approachshows that in the limit, $R^{tr}$ coincides with $R^F$. This suggests acomparison between the probabilistic interpretations of $R^F$ vs. $R^W$.

Author: Palle E. T. Jorgensen, Erin P. J. Pearse

Source: https://arxiv.org/